Passivity-based Stabilization of Non-Compact Sets - Systems Control ...
Passivity-based Stabilization of Non-Compact Sets Mohamed I. El-Hawwary and Manfredi Maggiore
Abstract— We investigate the stabilization of closed sets for passive nonlinear systems which are contained in the zero level set of the storage function.
I. I NTRODUCTION Equilibrium stabilization is one of the basic control specifications. It is one of the important research problems in nonlinear control theory which continues to receive much attention. The more general problem of stabilizing sets has received comparatively less attention. This problem has intrinsic interest because many control specifications can be naturally formulated as set stabilization requirements. The synchronization or state agreement problem, which entails making the states of two or more dynamical systems converge to each other, can be formulated as the problem of stabilizing the diagonal subspace in the state space of the coupled system. The observer design problem can be viewed in the same manner. The control of oscillations in a dynamical system can be thought of as the stabilization of a set homeomorphic to the unit circle or, more generally, to the k-torus. The maneuver regulation or path following problem, which entails making the output of a dynamical system approach and follow a specified path in the output space of a control system, can be thought of as the stabilization of a certain invariant subset of the state space compatible with the motion on the path. Recently, significant progress has been made toward a Lyapunov characterization of set stabilizability. Albertini and Sontag showed in [1] that uniform asymptotic controllability to a closed, possibly non-compact set is equivalent to the existence of a continuous control-Lyapunov function. Kellet and Teel in [2], [3], and [4], proved that for a locally Lipschitz control system, uniform global asymptotic controllability to a closed, possibly non-compact set is equivalent to the existence of a locally Lipschitz control Lyapunov function. Moreover, they were able to use this result to construct a semiglobal practical asymptotic stabilizing feedback. A geometric approach to a specific set stabilization problem for single-input systems was taken by Banaszuk and Hauser in [5]. There, the authors characterized conditions for dynamics transversal to an open-loop invariant periodic orbit to be feedback linearizable. In [6], Nielsen and one of the authors generalized Banaszuk and Hauser’s results This research was supported by the National Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada. {melhawwary,
maggiore}@control.utoronto.ca
to more general controlled-invariant sets. In [7], the same authors extended the theory to multi-input systems. In a series of papers, [8], [9], [10], Shiriaev and co-workers addressed the problem of stabilizing compact invariant sets for passive nonlinear systems. Their work can be seen as a direct extension of the equilibrium stabilization results for passive systems by Byrnes, Isidori, and Willems in [11]. One of the key ingredients is the notion of V -detectability which generalizes zero-state detectability introduced in [11]. This paper follows Shiriaev’s line of work by investigating the passivity-based stabilization of closed, possibly noncompact sets. Our setting is more general than Shiriaev’s in that, rather than requiring the goal set to be the zero level set of the storage function, we only require it to be a subset thereof. Moreover, the goal set may be unbounded. Our main contributions are as follows. •
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•
•
We introduce a more general notion of detectability with respect to sets, called Γ-detectability (Definition III.1), which is closer in spirit to the original notion of zero-state detectability. We provide geometric sufficient conditions to characterize Γ-detectability. We show (Theorem III.1) that Γ-detectability, among other conditions, guarantees that a passivity-based feedback renders the goal set attractive in the special case when all trajectories are bounded. In the case of unbounded trajectories, we provide sufficient conditions for a passivity-based feedback to render the goal set attractive or asymptotically stable (Theorems V.1 and V.2). We present necessary and sufficient conditions for the equivalence between output convergence and set attractivity (Theorem IV.1). II. P RELIMINARIES
AND
P ROBLEM S TATEMENT
In the sequel, we denote by φ(t, x0 , u(t)) the unique solution to a smooth differential equation x˙ = f (x) + g(x)u(t), with initial condition x0 and piecewise continuous control input signal u(t). Given a control input signal u(t), we let L+ (x0 , u(t)) and L− (x0 , u(t)) denote the positive and negative limit sets (ω and α limit sets) of φ(t, x0 , u(t)). Given a closed nonempty set Γ ⊂ X , where X is a vector space, a point ξ ∈ X , and a vector norm k · k : X → R, the point-toset distance kξkΓ is defined as kξkΓ := inf η∈Γ kξ − ηk. We use the standard notation Lf V to denote the Lie derivative of a C 1 function V along a vector field f , and dV (x) to denote the differential map of V . We denote by adf g the Lie bracket of two vector fields f and g, and by adkf g its k-th iteration.
In this paper, we consider the control-affine system, x˙ = f (x) + g(x)u y = h(x)
(1)
with state space X = Rn , set of input values U = Rm and set of output values Y = Rm . Throughout this paper, it is assumed that f and the m columns of g are smooth vector fields, and that h is a smooth mapping. It is further assumed that (1) is passive with smooth storage function V : X → R, i.e., V is a C r (r ≥ 1) nonnegative function such that, for all piecewise-continuous functions u : [0, ∞) → U, for all x0 ∈ X , and for all t in the maximal interval of existence of φ(·, x0 , u), Z t V (x(t)) − V (x0 ) ≤ u(τ )⊤ y(τ )dτ, (2) 0
where x(t) = φ(t, x0 , u(t)) and y(t) = h(x(t)). It is wellknown that the passivity property is equivalent to the two conditions (∀x ∈ X ) Lf V (x) ≤ 0 (∀x ∈ X ) Lg V (x) = h(x)⊤ . We next present stability definitions used in this paper. Let Γ ⊂ X be a closed invariant set for a system Σ : x˙ = f (x), x ∈ X. Definition II.1 (Set Stability, [12]) (i) Γ is stable with respect to Σ if for all ε > 0 there exists a neighbourhood N (Γ) of Γ, such that x0 ∈ N (Γ) ⇒ kx(t)kΓ < ε for all t ≥ 0. (ii) Γ is uniformly stable with respect to Σ if (∀ε > 0)(∃δ > 0)(∀x0 ∈ X ) (kx0 kΓ < δ ⇒ (∀t ≥ 0)kx(t)kΓ < ε). (iii) Γ is a semi-attractor of Σ if there exists a neighbourhood N (Γ) such that x0 ∈ N (Γ) ⇒ limt→∞ kx(t)kΓ = 0. (iv) Γ is an attractor of Σ if (∃δ > 0) (∀x0 ∈ X ) kx0 kΓ < δ ⇒ limt→∞ kx(t)kΓ = 0. (v) Γ is a global attractor of Σ if it is an attractor with δ = ∞ or a semi-attractor with N (Γ) = X . (vi) Γ is a stable semi-attractor of Σ if it is stable and semiattractive with respect to Σ. (vii) Γ is asymptotically stable with respect to Σ if it is a uniformly stable attractor of Σ. (viii) Γ is globally asymptotically stable with respect to Σ if it is a uniformly stable global attractor of Σ.
Problem Statement Given a closed set Γ ⊆ V −1 (0) = {x ∈ X : V (x) = 0} which is open-loop invariant for (1), we seek to find conditions under which a passivity-based feedback u = −ϕ(y) makes Γ either a stable semi-attractor, a (global) attractor, or a (globally) asymptotically stable set for the closed-loop system. We refer to Γ as the goal set. This problem was investigated by Shiriaev and co-workers in [8], [9], [10] in the case when Γ is compact and Γ = V −1 (0). Their work extended the landmark results by Byrnes, Isidori, and Willems in [11] developed for the case when Γ = V −1 (0) = {0}. We next review Shiriaev’s main results. Definition II.3 (V -detectability [8]-[9]) System (1) is locally V -detectable if there exists a constant c > 0 such that for all x0 ∈ Vc = {x ∈ X : V (x) ≤ c}, the solution x(t) = φ(t, x0 , 0) of the open-loop system satisfies y(t) = 0 ∀t ≥ 0 ⇒ limt→∞ V (x(t)) = 0. If c = ∞, then the system is V -detectable. Theorem II.1 ((Global) Asymptotic Stability [9]) Let ϕ : Y → U be any smooth function such that ϕ(0) = 0 and y ⊤ ϕ(y) > 0 for y 6= 0. Consider the feedback u = −ϕ(y). (i) If the set V −1 (0) is compact and the system is locally V -detectable, then V −1 (0) is asymptotically stable for (1). (ii) If the function V is proper and system (1) is V detectable, then V −1 (0) is globally asymptotically stable for (1). Shiriaev gave a sufficient condition for V -detectability which extends that of Proposition 3.4 in [11] for zero-state detectability. Proposition II.1 (Condition for V -detectability [8]-[9]) Let S = {x ∈ X : Lm f Lτ V (x) = 0, ∀τ ∈ D, 0 ≤ m < r}, k where D is the distribution D = span{ad S f gi : 0+≤ k ≤ n − 1, 1 ≤ i ≤ m}. Further, define Ω = x0 ∈X L (x0 , 0). If S ∩ Ω ⊂ V −1 (0) then system (1) is V -detectable. III. S ET S TABILIZATION - C ASE
OF BOUNDED
TRAJECTORIES
Remark II.1 If Γ is compact, then Γ is stable if, and only if, it is uniformly stable. Moreover, Γ is a semi-attractor if, and only if, it is an attractor.
The notion of V -detectability in Definition II.3 is only applicable to the situation when Γ = V −1 (0). If Γ ( V −1 (0), then the property x(t) → V −1 (0) does not imply that x(t) → Γ. For this reason, we introduce a different generalization of the zero-state detectability notion which is independent of the storage function and, as such, is closer in spirit to the original definition of zero-state detectability in [11].
Definition II.2 (Relative Set Stability) Let Ξ ⊂ X be such that Ξ ∩ Γ 6= ∅. Γ is stable with respect to Σ relative to Ξ if, for any ε > 0, there exists a neighbourhood N (Γ) such that x0 ∈ N (Γ) ∩ Ξ ⇒ kx(t)kΓ < ε for all t ≥ 0. Similarly, one modifies all other notions in Definition II.1 by restricting initial conditions to lie in Ξ.
Definition III.1 (Γ-Detectability) System (1) is locally Γdetectable if there exists a neighborhood N (Γ) of Γ such that, for all x0 ∈ N (Γ), the solution x(t) = φ(t, x0 , 0) satisfies y(t) = 0 ∀t ≥ 0 ⇒ limt→∞ kx(t)kΓ = 0. The system is Γ-detectable if it is locally Γ-detectable with N (Γ) = X .
ω−Limit Sets
Remark III.1 If Γ = V −1 (0) and V is proper (and therefore Γ is compact), then system (1) is (locally) Γ-detectable if, and only if, it is (locally) V -detectable.
4 3 2
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The sufficient condition for zero-state detectability of Proposition 3.4 in [11] is easily extended to get a condition for Γ-detectability as follows.
5
0 −1 −2
Proposition III.1 (Condition for Γ-detectability) Let S and D be defined as in Proposition II.1. Suppose that all trajectories on the maximal open-loop invariant subset of h−1 (0) are bounded. If S ∩ Ω ⊂ Γ then system (1) is Γ-detectable.
−3 −4 −5 −5
−3
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x1 Initial State = (1, 1, 1)
The proof of this proposition is a straightforward extention of that of Proposition 3.4 in [11], and is therefore omitted.
1 0.8
x3
Remark III.2 The assumption, in [11] and [8]-[9], that V is proper implies that all trajectories of the open-loop system are bounded.
−4
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The next, equivalent, condition for Γ-detectability does not require the knowledge of the storage function V .
0 2 1
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Proposition III.2 Let S and Ω be defined as in Proposition II.1 and define S ′ = {x ∈ X : Lm f h(x) = 0, 0 ≤ m ≤ r + n − 2}. Then, S ∩ Ω = S ′ ∩ Ω and therefore under the assumption of Proposition III.1 system (1) is Γ-detectable if S ′ ∩ Ω ⊂ Γ. The proof of the this proposition is omitted for brevity. Shiriaev’s result in Theorem II.1 states that V detectability, and hence Γ-detectability, is a sufficient condition to asymptotically stabilize Γ in the case when Γ = V −1 (0) and Γ is compact. In the more general setting under investigation, that is, when Γ ⊂ V −1 (0) and Γ is not necessarily compact, Γ-detectability is no longer a sufficient condition for asymptotic stability of Γ. We show this by means of the following counterexample. Example III.1 Consider the following system: x˙ 1 = (x22 + x23 )(−x2 ), x˙ 2 = (x22 + x23 )(x1 ), x˙ 3 = −x33 + u, with output function h(x1 , x2 , x3 ) = x33 . This system is passive with storage V = 14 x43 . On the set h−1 (0) = {(x1 , x2 , x3 ) : x3 = 0}, the system dynamics take the form: x˙ 1 = (x22 )(−x2 ), x˙ 2 = (x22 )(x1 ). If the goal set is given as Γ = {(x1 , x2 , x3 ) : x2 = x3 = 0}, then it can be seen from Figure 1 that all trajectories originating in h−1 (0) approach Γ and hence the system is Γ-detectable. Using the control u = −y, all system trajectories are bounded for all x0 ∈ R3 . Thus, for all x0 ∈ R3 , the positive limit set L+ (x0 , u) is nonempty, compact, and formed by trajectories on h−1 (0) = {x3 = 0}. For x0 ∈ / (h−1 (0) ∪ {(x1 , x2 , x3 ) : x1 = x2 = 0)}), it can be shown that the positive limit set L+ (x0 , u) is a circle which intersects Γ at equilibrium points. Thus, x(t) 6→ Γ because L+ (x0 , 0) 6⊂ Γ. This is illustrated in Figure 1. In this example the feedback u = −y guarantees that all system trajectories are bounded for all initial conditions, and
0
−1
x2
−2
Fig. 1.
−1 −2
x1
Counterexample
x(t) → h−1 (0) = V −1 (0). The Γ-detectability property guarantees that on V −1 (0), which is open-loop invariant, all trajectories approach Γ. However, these two facts do not imply that x(t) → Γ. The problem lies in the fact that, while Γ is a global attractor relative to V −1 (0), Γ is not stable relative to V −1 (0). We next introduce sufficient conditions to insure that Γ is a global attractor of the closed-loop system in the special case when all trajectories of the system are bounded. The condition, in the result below, dealing with relative stability of Γ is similar to a condition presented in [13] for equilibrium stabilization of positive systems with first integrals. Part of the proof of our result is inspired by the stability results using positive semi-definite Lyapunov functions presented in [14]. Theorem III.1 Consider system (1) and a feedback u = −ϕ(y), where ϕ : Y → U is a smooth function such that ϕ(0) = 0 and y ⊤ ϕ(y) > 0 for y 6= 0. Suppose that, for all x0 ∈ X [resp., for all x0 in a neighborhood of Γ], x(t) = φ(t, x0 , u) is bounded. If the system is Γ-detectable [resp., locally Γ-detectable] and Γ is stable with respect to the openloop system relative to V −1 (0), then Γ is a global attractor [resp., a semi-attractor] of the closed-loop system. Remark III.3 When Γ = V −1 (0), then the relative stability assumption of Theorem III.1 is trivially fulfilled.
Example III.2 Consider the following system: x˙ 1 = x2 x3 , x˙ 2 = x1 x3 − x32 , x˙ 3 = −x3 x42 + u, with output y = x3 . This system is passive with positive semi-definite storage V (x) = 1 2 2 2 x3 . Indeed, Lg V (x) = x3 and Lf V (x) = −x3 x2 ≤ 0. Consider the goal set Γ = {x : x3 = x2 = 0}. The set V −1 (0) = h−1 (0) = {x : x3 = 0} is open-loop invariant and the dynamics of the system on the set are given by x˙ 1 = 0, x˙ 2 = −x32 . Clearly, the system is Γ detectable and Γ is stable with respect to the open-loop system relative to V −1 (0). The feedback u = −y globally stabilizes Γ. IV. O UTPUT C ONVERGENCE
VS .
S ET ATTRACTIVITY
When the goal set Γ can be expressed as Γ = γ −1 (0) = {x ∈ X : γ(x) = 0}, where γ is a continuous function X → Rp , one approach to stabilizing Γ is to view γ(x) as the output of the system and pose the problem of stabilizing Γ as that of stabilizing the output. In general, however, the convergence of the output to zero is not equivalent to the
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(a) γ(x(t)) → 0 6⇒ x(t) → γ −1 (0) 2 1.5 1
level sets of γ
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Proof: We prove the global result of the theorem. The proof of the local result is very similar mutatis mutandis. Since the trajectory x(t) = φ(t, x0 , u) is bounded, its positive limit set, L+ (x0 , u), is nonempty, compact, and invariant for the closed-loop system. R t Along the trajectory x(t) we have V (x(t)) − V (x0 ) ≤ 0 −y(τ )⊤ ϕ(y(τ ))dτ ≤ 0. Since V is continuous and nonincreasing, limt→∞ V (x(t)) = c ≥ 0. This implies that V (x) = c on L+ (x0 , u). Let x ¯ be a point of L+ (x0 , u) and set x ¯(t) = φ(t, x ¯, u). Since x¯(t) ∈ L+ (x0 , u), V (¯ x(t)) = c for all t ≥ 0 and 0 = V (¯ x(t)) − V (¯ x) ≤ Rthen t ⊤ −y(τ ) ϕ(y(τ ))dτ ≤ 0, so that y(t) = h(¯ x (t)) = 0, ∀t ≥ 0 0. By Γ-detectability, x ¯(t) → Γ ⊂ V −1 (0), and hence c = 0. We have thus shown that L+ (x0 , u) ⊂ V −1 (0). In order to show that x(t) → Γ we will show that, by the assumption of stability of Γ relative to V −1 (0), L+ (x0 , u) ⊂ Γ. Assume, by way of contradiction, that L+ (x0 , u) 6⊂ Γ, so that we can find a point p ∈ L+ (x0 , u) and p ∈ / Γ. Since V is positive semi-definite, it follows that for all x ∈ V −1 (0), dV (x) = 0, and hence Lg V (x) = 0, proving that V −1 (0) ⊂ h−1 (0). This fact, the invariance of L+ (x0 , u), and the fact that L+ (x0 , u) ⊂ V −1 (0) together imply that φ(t, p, u) = φ(t, p, 0) for all t ∈ R, and therefore φ(t, p, 0) ∈ L+ (x0 , u) for all t ∈ R. The invariance and closedness of L+ (x0 , u) also imply that L− (p, 0) ⊂ L+ (x0 , u) ⊂ V −1 (0). Since L− (p, 0) ⊂ V −1 (0) is open-loop invariant, by the Γ-detectability property we have that open-loop trajectories originating in L− (p, 0) are contained in L− (p, 0) and approach Γ in positive time; this fact and the closedness of L− (p, 0) imply that L− (p, 0) ∩ Γ 6= ∅. Let x ¯ be a point in L− (p, 0) ∩ Γ. Since Γ is stable with respect to the openloop system relative to V −1 (0), for any ǫ > 0 there exists a neighborhood N (Γ) such that for all z ∈ N (Γ) ∩ V −1 (0), kφ(t, z, 0)kΓ < ǫ for all t ≥ 0. Pick ǫ > 0 such that kpkΓ > ǫ. Since x¯ ∈ Γ is a negative limit point of φ(t, p, 0), there exist a point z ∈ N (Γ) and a time T > 0 such that p = φ(T, z, 0), contradicting the relative stability of Γ. Hence, L+ (x0 , u) ⊂ Γ. Since φ(t, x0 , u) → L+ (x0 , u), we conclude that φ(t, x0 , u) → Γ.
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(b) x(t) → γ −1 (0) 6⇒ γ(x(t)) → 0 Fig. 2. Counterexamples showing that output convergence is not equivalent to set attractivity.
attractivity of the goal set. This is illustrated in the next two counterexamples. Example IV.1 (γ(x(t)) → 0 6⇒ x(t) → γ −1 (0)) Consider −x2 the system x˙ 1 = 1, x˙ 2 = 1+x with output function 2 1 x2 γ(x1 , x2 ) = 1+x2 . The zero level set of this function 1 is given by γ −1 (0) = {x : x2 = 0}. The solution with initial condition (x10 , x20 ) is x1 (t) = x10 + t, x2 (t) = x20 exp(− arctan(x10 + t) + arctan x10 ). It is clear that if x20 6= 0, then x(t) 6→ γ −1 (0). The output signal is γ(x(t)) = [x20 exp(− arctan(x10 + t) + arctan x10 )]/[1 + (x10 + t)2 ]. Clearly, γ(x(t)) → 0. The problem here is that for all c 6= 0, the distance between points in γ −1 (c) and the set γ −1 (0) is not uniformly bounded; see Figure 2(a). Example IV.2 (x(t) → γ −1 (0) 6⇒ γ(x(t)) → 0) Consider 1 x2 with output function the system x˙ 1 = 1, x˙ 2 = −x 1+x21 2 2 γ(x1 (t), x2 (t)) = x2 (1 + x1 ). The zero level set of this function is the set γ −1 (0) = {x : x2 = 0}. The solution with initial condition (x10 , x20 ) is x1 (t) = x10 + t,
p x2 (t) = x20 (1 + x210 )/(t2 + 2x10 t + 1 + x210 ), so for all initial conditions, x(t) → γ −1 (0). On the other hand, γ(x1 (t), x2 (t)) = x220 [(1 + x210 )[(x10 + t)2 + 1]]/[t2 + 2x10 t + 1 + x210 ], so if x20 6= 0, γ(x1 (t), x2 (t)) 6→ 0. The problem here is that for all c 6= 0, the distance between the sets γ −1 (c) and γ −1 (0) is not bounded away from zero; see Figure 2(b). We next present necessary and sufficient conditions to establish the equivalence between output convergence and the attractivity of γ −1 (0). The proof is omitted due to space limitations. Theorem IV.1 Let γ : X → Rp be a continuous function, and let Γ = {x ∈ X : γ(x) = 0}. (i) (Class K-lower bound) A necessary and sufficient condition such that (∀{xn }n∈N ⊂ X ) (γ(xn ) → 0 =⇒ xn → Γ), is that there exists r > 0 and a class-K function α : [0, r) → R+ such that (∀kxkΓ < r) α(kxkΓ ) ≤ kγ(x)k. (ii) (Class K-upper bound) A necessary and sufficient condition such that (∀{xn }n∈N ⊂ X ) (xn → Γ =⇒ γ(xn ) → 0), is that there exists r > 0 and a class-K function β : [0, r) → R+ such that (∀kxkΓ < r) kγ(x)k ≤ β(kxkΓ ). (iii) (Bounded Sequences) For all bounded sequences {xn }n∈N ⊂ X , γ(xn ) → 0 ⇐⇒ xn → Γ. V. S ET S TABILIZATION - C ASE
OF UNBOUNDED
TRAJECTORIES
We now turn our attention to situation when trajectories are unbounded and present two results concerning the (semi) attractivity and asymptotic stability of Γ. A key ingredient here is the result in Theorem IV.1. Theorem V.1 Consider system (1) and a feedback u = −ϕ(x), where ϕ : X → U is a smooth function such that h(x)⊤ ϕ(x) ≥ ψ(kh(x)k), and ψ is a class-K function. Define λ(x) = h(x)⊤ ϕ(x). Suppose that the closed-loop system has no finite escape times and that: (i) There exists r > 0 and a class-K function α : [0, r) → R+ such that α(kxkΓ ) ≤ kh(x)k for all kxkΓ < r. (ii) There exists s ∈ (0, r] such that inf kxkΓ =s V (x) > 0. (iii) For all c > 0 there exists k ∈ R such that whenever V (x) ≤ c, either Lf λ(x) − Lg λ(x)ϕ(x) ≤ k or Lf λ(x) − Lg λ(x)ϕ(x) ≥ k. Then, Γ is a semi-attractor. If (i)-(iii) hold and (iv) there exists a class-K function β : [0, r) → R+ such that V (x) ≤ β(kxkΓ ) for all kxkΓ < r, then Γ is an attractor. Finally, if (i), (iii) hold and
(v) Γ = V −1 (0) = h−1 (0), then Γ is a global attractor. Remark V.1 Assumption (i) implies that Γ = V −1 (0)∩{x ∈ X : kxkΓ < r} = h−1 (0) ∩ {x ∈ X : kxkΓ < r}. In other words, the assumption imposes that each connected component of Γ coincides with a connected component of V −1 (0) and h−1 (0). Proof: Let x0 ∈ X be arbitrary and denote x(t) = φ(t, x0 , u). SinceR V (·) is nonnegative and t V (x(t)) − V (x(0)) ≤ − 0 λ(x(τ )) dτ ≤ 0, this implies that V (x(t)) ≤ V (x(0)) for all t ≥ 0 and Rt thus limt→∞ 0 λ(x(τ )) dτ exists and is finite. By (iii), ˙ there exists k ∈ R such that λ(x(t)) = Lf λ(x(t)) + Lg λ(x(t))ϕ(x(t)) is either ≤ k or ≥ k. Take the case where ˙ λ(x(t)) ≥ k; we will show that λ(x(t)) → 0. Suppose, by way of contradition, that λ(x(t)) 6→ 0. This implies that there exists ǫ > 0 and a sequence {tn } such that tn → ∞ and λ(x(tn )) ≥ ǫ for all n. By the mean value theorem, for all n and for all t ≥ tn , there exists sn ∈ [tn , t] such that ˙ λ(x(t)) = λ(x(tn )) + λ(x(s n ))(t − tn ) ≥ ǫ + k(t − tn ) ≥ ǫ − |k| |t − tn |. Pick ǫ0 ∈ (0, ǫ) and let δ = (ǫ − ǫ0 )/|k|. Then, for all n and for all t ∈ [tn , tn + δ], λ(t) ≥ ǫ0 > 0. Take a subsequence {tnk } such that tnk+1 − tnk ≥ δ. Since, by assumption, λ(x(t)) ≥ 0, we have Z t X X Z tnk +δ λ(x(τ ))dτ ≥ λ(x(τ ))dτ ≥ lim δǫ0 , t→∞
0
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k
Rt
contradicting the fact that limt→∞ 0 λ(x(τ )) dτ exists and is finite. Thus, λ(x(t)) → 0 and since ψ(kh(x)k) ≤ λ(x(t)), ˙ h(x(t)) → 0. The proof for the case λ(x(t)) ≤ k is almost identical and therefore omitted. Recall (see the proof of Theorem III.1) that Γ ⊂ V −1 (0) ⊂ h−1 (0). By assumption (i), Γ = V −1 (0) ∩ {kxkΓ < r} = h−1 (0) ∩ {kxkΓ < r}. Moreover, for all kxkΓ < r, kh(x)k ≥ α(kxkΓ ) ≥ α(kxkh−1 (0) ). By (ii), c = inf kxkΓ =s V (x) is positive. Let N (Γ) = {x ∈ X : kxk < s, V (x) < c}. It follows that N (Γ) is positively invariant and thus for all x0 ∈ N (Γ), h(x(t)) → 0 and kx(t)kΓ < r. This, together with the fact that kh(x(t))k ≥ α(kx(t)kh−1 (0) ) implies that x(t) → h−1 (0)∩{kxkΓ < r} = Γ, proving that Γ is a semi-attractor. If, in addition to assumptions (i)-(iii), assumption (iv) holds then there exists δ > 0 such that {x ∈ X : kxkΓ < δ} ⊂ N (Γ)}, and thus Γ is an attractor. If assumptions (i), (iii) and (v) hold, then we have shown that (∀x0 ∈ X ) h(φ(t, x0 , u)) → 0 and by Theorem IV.1, φ(t, x0 , u) → h−1 (0) = Γ. Thus, Γ is a global attractor. By strengthening conditions (i), (ii) and removing condition (iii) in Theorem V.1, we gain stability. Theorem V.2 Consider system (1) and a feedback u = −ϕ(x), where ϕ : X → U is a smooth function such that ϕ(x)⊤ h(x) ≥ ψ(kh(x)k), and ψ is a class-K function. Suppose that the closed-loop system has no finite escape times and that:
(i) There exists r > 0 and a class-K function α : [0, r) → R+ such that α(kxkΓ ) ≤ V (x) for all kxkΓ < r. (ii) There exists a class-K function µ : [0, +∞) → R+ such that µ(V (x)) ≤ kh(x)k for all x ∈ X . Then, Γ is a stable semi-attractor of the closed-loop system. If (i), (ii) hold and (iii) there exists a class-K function β : [0, r) → R+ such that V (x) ≤ β(kxkΓ ) for all kxkΓ < r, then Γ is asymptotically stable. If (i), (ii) hold and (iv) Γ = V −1 (0) = h−1 (0), then Γ is globally attractive. Finally, if (i)-(iv) hold, then Γ is globally asymptotically stable. Remark V.2 Assuptions (i), (ii) in Theorem V.2 imply assumptions (i), (ii) of Theorem V.1. Indeed, for all kxkΓ < r, kh(x)k ≥ µ(V (x)) ≥ µ ◦ α(kxkΓ ). Moreover, the inequality V (x) ≥ α(kxkΓ ) implies inf kxkΓ =s V (x) ≥ α(s) > 0 for all s ∈ (0, r]. Proof: By assumption (ii), for all x0 ∈ X , dV (φ(t, x0 , u)) ≤ −h(φ(t, x0 , u))⊤ ϕ(φ(t, x0 , u)) dt ≤ −ψ ◦ µ(V (φ(t, x0 , u))). By the comparison lemma, it follows that V (φ(t, x0 , u)) → 0. Let ǫ ∈ (0, r), choose c > 0 such that α−1 (c) < ǫ, and define N (Γ) = {x ∈ X : V (x) < c, kxkΓ < r}. Clearly, N (Γ) is a neigbourhood of Γ. By our choice of c and assumption (i), N (Γ) ⊂ {x ∈ X : kxkΓ < ǫ) and the boundary of N (Γ) is ∂N (Γ) = {x ∈ X : V (x) = c, kxkΓ < r}. Since V is nonincreasing, it follows that N (Γ) is positively invariant and hence Γ is stable. Since V (φ(t, x0 , u)) → 0 for all x0 ∈ N (Γ), assumption (i) implies that x(t) → Γ. Hence, Γ is a semi-attractor. If, in addition, assumption (iii) holds, then there exists δ > 0 such that {x ∈ X : kxkΓ < δ} ⊂ N (Γ), and thus Γ is asymptotically stable. If assumptions (i), (ii), and (iv) hold, then we have shown that (∀x0 ∈ X ) V (φ(t, x0 , u)) → 0. Hence, by Theorem IV.1, φ(t, x0 , u) → V −1 (0) = Γ. Thus, Γ is globally attractive. If assumptions (i)-(iv) hold, then Γ is a uniformly stable global attractor, that is, Γ is globally asymptotically stable.
Example V.1 Consider the system x˙ 1 = tan−1 (x1 ), x˙ 2 = x2 (1+x21 )u, with output y = x22 . This system is passive with x22 a storage function V (x) = 12 1+x 2 . Let the goal set be Γ = 1 V −1 (0) = h−1 (0) = {x : x2 = 0}. It is seen that V does not have a class-K lower bound with respect to Γ and thus Theorem V.2 cannot be applied to this example. On the other hand, h(x) does have a class-K lower bound with respect to Γ, as kh(x)k = kxk2Γ . Moreover, kh(x)k = kxk2h−1 (0) . Using the control u = −ϕ(x), with ϕ(x) = h(x), we have Lf λ(x) − Lg λ(x)ϕ(x) = −4x62 (1 + x21 ), which is bounded
from above. Thus we have that conditions (i), (iii) and (v) of Theorem V.1 are satisfied and Γ is globally attractive. Example V.2 Consider the system x˙ 1 = 1, x˙ 2 = (1 + 0.5 cos(x21 ))u, with output y = x2 (1+0.5 cos(x21 )), which is passive with storage function V (x) = 21 x22 . Let the goal set be Γ = {x : x2 = 0} so that V −1 (0) = h−1 (0) = Γ. Moreover, V (x) = 1/2kxk2Γ and kh(x)k ≥ 0.5|x2 | = 0.5kxkΓ , so by Theorem V.2 the feedback u = −ϕ(x), with ϕ(x) = h(x), renders Γ globally asymptotically stable. It is easily seen that Theorem V.1 cannot be applied to this example because assumption (iii) does not hold. VI. C ONCLUSIONS This paper leaves several open questions that we will investigate in future work. In Theorem III.1, we assume that the trajectories of the closed-loop system are bounded. Boundedness is a property that one should seek to achieve by means of a suitable feedback. The same comment holds for Theorems V.1 and V.2, where we assume that the closed-loop system does not possess finite escape times. R EFERENCES [1] F. Albertini and E. Sontag, “Continuous control-Lyapunov functions for asymptotically controllable time-varying systems,” International Journal of. Control, vol. 72, pp. 1630–1641, 1999. [2] C. M. Kellet and A. R. Teel, “Asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function,” in Proc. of the 39th Conference on Decision and Control, (Sydney, Australia), 2000. [3] C. M. Kellet and A. R. Teel, “A converse Lyapunov theorem for weak uniform asymptotic controllability of sets,” in Proc. of the 14th MTNS, (France), 2000. [4] C. M. Kellet and A. R. Teel, “Weak converse Lyapunov theorems and control-Lyapunov functions,” SIAM Journal on Control and Optimization, vol. 42, no. 6, pp. 1934–1959, 2004. [5] A. Banaszuk and J. Hauser, “Feedback linearization of transverse dynamics for periodic orbits,” Systems and Control Letters, vol. 29, pp. 95–105, 1995. [6] C. Nielsen and M. Maggiore, “Output stabilization and maneuver regulation: A geometric approach,” Systems and Control Letters, vol. 55, pp. 418–427, 2006. [7] C. Nielsen and M. Maggiore, “Further results on transverse feedback linearization of multi-input systems,” in Proc. of the 45th IEEE Conference on Decision and Control, (San Diego, CA, USA), December 2006. [8] A. Shiriaev, “Stabilization of compact sets for passive affine nonlinear systems,” Automatica, vol. 36, pp. 1373–1379, 2000. [9] A. Shiriaev, “The notion of V -detectability and stabilization of invariant sets of nonlinear systems,” Systems and Control Letters, vol. 39, pp. 327–338, 2000. [10] A. Shiriaev and A. L. Fradkov, “Stabilization of invariant sets for nonlinear non-affine systems,” Automatica, vol. 36, pp. 1709–1715, 2000. [11] C. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence, and the global stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 36, pp. 1228–1240, 1991. [12] N. P. Bathia and G. P. Szeg¨o, Dynamical Systems: Stability Theory and Applications. Berlin: Springer-Verlag, 1967. [13] P. de Leenheer and D. Aeyels, “Stabilization of positive systems with first integrals,” Automatica, vol. 38, pp. 1583 – 1589, 2002. [14] A. Iggidr, B. Kalitine, and R. Outbib, “Semidefinite Lyapunov functions stability and stabilization,” Mathematics of Control, Signals and Systems, vol. 9, pp. 95–106, 1996.
Abstract— We investigate the stabilization of closed sets for passive nonlinear systems which are contained in the zero level set of the storage function.
I. I NTRODUCTION Equilibrium stabilization is one of the basic control specifications. It is one of the important research problems in nonlinear control theory which continues to receive much attention. The more general problem of stabilizing sets has received comparatively less attention. This problem has intrinsic interest because many control specifications can be naturally formulated as set stabilization requirements. The synchronization or state agreement problem, which entails making the states of two or more dynamical systems converge to each other, can be formulated as the problem of stabilizing the diagonal subspace in the state space of the coupled system. The observer design problem can be viewed in the same manner. The control of oscillations in a dynamical system can be thought of as the stabilization of a set homeomorphic to the unit circle or, more generally, to the k-torus. The maneuver regulation or path following problem, which entails making the output of a dynamical system approach and follow a specified path in the output space of a control system, can be thought of as the stabilization of a certain invariant subset of the state space compatible with the motion on the path. Recently, significant progress has been made toward a Lyapunov characterization of set stabilizability. Albertini and Sontag showed in [1] that uniform asymptotic controllability to a closed, possibly non-compact set is equivalent to the existence of a continuous control-Lyapunov function. Kellet and Teel in [2], [3], and [4], proved that for a locally Lipschitz control system, uniform global asymptotic controllability to a closed, possibly non-compact set is equivalent to the existence of a locally Lipschitz control Lyapunov function. Moreover, they were able to use this result to construct a semiglobal practical asymptotic stabilizing feedback. A geometric approach to a specific set stabilization problem for single-input systems was taken by Banaszuk and Hauser in [5]. There, the authors characterized conditions for dynamics transversal to an open-loop invariant periodic orbit to be feedback linearizable. In [6], Nielsen and one of the authors generalized Banaszuk and Hauser’s results This research was supported by the National Sciences and Engineering Research Council of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, ON, M5S 3G4, Canada. {melhawwary,
maggiore}@control.utoronto.ca
to more general controlled-invariant sets. In [7], the same authors extended the theory to multi-input systems. In a series of papers, [8], [9], [10], Shiriaev and co-workers addressed the problem of stabilizing compact invariant sets for passive nonlinear systems. Their work can be seen as a direct extension of the equilibrium stabilization results for passive systems by Byrnes, Isidori, and Willems in [11]. One of the key ingredients is the notion of V -detectability which generalizes zero-state detectability introduced in [11]. This paper follows Shiriaev’s line of work by investigating the passivity-based stabilization of closed, possibly noncompact sets. Our setting is more general than Shiriaev’s in that, rather than requiring the goal set to be the zero level set of the storage function, we only require it to be a subset thereof. Moreover, the goal set may be unbounded. Our main contributions are as follows. •
•
•
•
We introduce a more general notion of detectability with respect to sets, called Γ-detectability (Definition III.1), which is closer in spirit to the original notion of zero-state detectability. We provide geometric sufficient conditions to characterize Γ-detectability. We show (Theorem III.1) that Γ-detectability, among other conditions, guarantees that a passivity-based feedback renders the goal set attractive in the special case when all trajectories are bounded. In the case of unbounded trajectories, we provide sufficient conditions for a passivity-based feedback to render the goal set attractive or asymptotically stable (Theorems V.1 and V.2). We present necessary and sufficient conditions for the equivalence between output convergence and set attractivity (Theorem IV.1). II. P RELIMINARIES
AND
P ROBLEM S TATEMENT
In the sequel, we denote by φ(t, x0 , u(t)) the unique solution to a smooth differential equation x˙ = f (x) + g(x)u(t), with initial condition x0 and piecewise continuous control input signal u(t). Given a control input signal u(t), we let L+ (x0 , u(t)) and L− (x0 , u(t)) denote the positive and negative limit sets (ω and α limit sets) of φ(t, x0 , u(t)). Given a closed nonempty set Γ ⊂ X , where X is a vector space, a point ξ ∈ X , and a vector norm k · k : X → R, the point-toset distance kξkΓ is defined as kξkΓ := inf η∈Γ kξ − ηk. We use the standard notation Lf V to denote the Lie derivative of a C 1 function V along a vector field f , and dV (x) to denote the differential map of V . We denote by adf g the Lie bracket of two vector fields f and g, and by adkf g its k-th iteration.
In this paper, we consider the control-affine system, x˙ = f (x) + g(x)u y = h(x)
(1)
with state space X = Rn , set of input values U = Rm and set of output values Y = Rm . Throughout this paper, it is assumed that f and the m columns of g are smooth vector fields, and that h is a smooth mapping. It is further assumed that (1) is passive with smooth storage function V : X → R, i.e., V is a C r (r ≥ 1) nonnegative function such that, for all piecewise-continuous functions u : [0, ∞) → U, for all x0 ∈ X , and for all t in the maximal interval of existence of φ(·, x0 , u), Z t V (x(t)) − V (x0 ) ≤ u(τ )⊤ y(τ )dτ, (2) 0
where x(t) = φ(t, x0 , u(t)) and y(t) = h(x(t)). It is wellknown that the passivity property is equivalent to the two conditions (∀x ∈ X ) Lf V (x) ≤ 0 (∀x ∈ X ) Lg V (x) = h(x)⊤ . We next present stability definitions used in this paper. Let Γ ⊂ X be a closed invariant set for a system Σ : x˙ = f (x), x ∈ X. Definition II.1 (Set Stability, [12]) (i) Γ is stable with respect to Σ if for all ε > 0 there exists a neighbourhood N (Γ) of Γ, such that x0 ∈ N (Γ) ⇒ kx(t)kΓ < ε for all t ≥ 0. (ii) Γ is uniformly stable with respect to Σ if (∀ε > 0)(∃δ > 0)(∀x0 ∈ X ) (kx0 kΓ < δ ⇒ (∀t ≥ 0)kx(t)kΓ < ε). (iii) Γ is a semi-attractor of Σ if there exists a neighbourhood N (Γ) such that x0 ∈ N (Γ) ⇒ limt→∞ kx(t)kΓ = 0. (iv) Γ is an attractor of Σ if (∃δ > 0) (∀x0 ∈ X ) kx0 kΓ < δ ⇒ limt→∞ kx(t)kΓ = 0. (v) Γ is a global attractor of Σ if it is an attractor with δ = ∞ or a semi-attractor with N (Γ) = X . (vi) Γ is a stable semi-attractor of Σ if it is stable and semiattractive with respect to Σ. (vii) Γ is asymptotically stable with respect to Σ if it is a uniformly stable attractor of Σ. (viii) Γ is globally asymptotically stable with respect to Σ if it is a uniformly stable global attractor of Σ.
Problem Statement Given a closed set Γ ⊆ V −1 (0) = {x ∈ X : V (x) = 0} which is open-loop invariant for (1), we seek to find conditions under which a passivity-based feedback u = −ϕ(y) makes Γ either a stable semi-attractor, a (global) attractor, or a (globally) asymptotically stable set for the closed-loop system. We refer to Γ as the goal set. This problem was investigated by Shiriaev and co-workers in [8], [9], [10] in the case when Γ is compact and Γ = V −1 (0). Their work extended the landmark results by Byrnes, Isidori, and Willems in [11] developed for the case when Γ = V −1 (0) = {0}. We next review Shiriaev’s main results. Definition II.3 (V -detectability [8]-[9]) System (1) is locally V -detectable if there exists a constant c > 0 such that for all x0 ∈ Vc = {x ∈ X : V (x) ≤ c}, the solution x(t) = φ(t, x0 , 0) of the open-loop system satisfies y(t) = 0 ∀t ≥ 0 ⇒ limt→∞ V (x(t)) = 0. If c = ∞, then the system is V -detectable. Theorem II.1 ((Global) Asymptotic Stability [9]) Let ϕ : Y → U be any smooth function such that ϕ(0) = 0 and y ⊤ ϕ(y) > 0 for y 6= 0. Consider the feedback u = −ϕ(y). (i) If the set V −1 (0) is compact and the system is locally V -detectable, then V −1 (0) is asymptotically stable for (1). (ii) If the function V is proper and system (1) is V detectable, then V −1 (0) is globally asymptotically stable for (1). Shiriaev gave a sufficient condition for V -detectability which extends that of Proposition 3.4 in [11] for zero-state detectability. Proposition II.1 (Condition for V -detectability [8]-[9]) Let S = {x ∈ X : Lm f Lτ V (x) = 0, ∀τ ∈ D, 0 ≤ m < r}, k where D is the distribution D = span{ad S f gi : 0+≤ k ≤ n − 1, 1 ≤ i ≤ m}. Further, define Ω = x0 ∈X L (x0 , 0). If S ∩ Ω ⊂ V −1 (0) then system (1) is V -detectable. III. S ET S TABILIZATION - C ASE
OF BOUNDED
TRAJECTORIES
Remark II.1 If Γ is compact, then Γ is stable if, and only if, it is uniformly stable. Moreover, Γ is a semi-attractor if, and only if, it is an attractor.
The notion of V -detectability in Definition II.3 is only applicable to the situation when Γ = V −1 (0). If Γ ( V −1 (0), then the property x(t) → V −1 (0) does not imply that x(t) → Γ. For this reason, we introduce a different generalization of the zero-state detectability notion which is independent of the storage function and, as such, is closer in spirit to the original definition of zero-state detectability in [11].
Definition II.2 (Relative Set Stability) Let Ξ ⊂ X be such that Ξ ∩ Γ 6= ∅. Γ is stable with respect to Σ relative to Ξ if, for any ε > 0, there exists a neighbourhood N (Γ) such that x0 ∈ N (Γ) ∩ Ξ ⇒ kx(t)kΓ < ε for all t ≥ 0. Similarly, one modifies all other notions in Definition II.1 by restricting initial conditions to lie in Ξ.
Definition III.1 (Γ-Detectability) System (1) is locally Γdetectable if there exists a neighborhood N (Γ) of Γ such that, for all x0 ∈ N (Γ), the solution x(t) = φ(t, x0 , 0) satisfies y(t) = 0 ∀t ≥ 0 ⇒ limt→∞ kx(t)kΓ = 0. The system is Γ-detectable if it is locally Γ-detectable with N (Γ) = X .
ω−Limit Sets
Remark III.1 If Γ = V −1 (0) and V is proper (and therefore Γ is compact), then system (1) is (locally) Γ-detectable if, and only if, it is (locally) V -detectable.
4 3 2
Γ
1
x2
The sufficient condition for zero-state detectability of Proposition 3.4 in [11] is easily extended to get a condition for Γ-detectability as follows.
5
0 −1 −2
Proposition III.1 (Condition for Γ-detectability) Let S and D be defined as in Proposition II.1. Suppose that all trajectories on the maximal open-loop invariant subset of h−1 (0) are bounded. If S ∩ Ω ⊂ Γ then system (1) is Γ-detectable.
−3 −4 −5 −5
−3
−2
−1
0
1
2
3
4
5
x1 Initial State = (1, 1, 1)
The proof of this proposition is a straightforward extention of that of Proposition 3.4 in [11], and is therefore omitted.
1 0.8
x3
Remark III.2 The assumption, in [11] and [8]-[9], that V is proper implies that all trajectories of the open-loop system are bounded.
−4
0.6 Γ
0.4 0.2
The next, equivalent, condition for Γ-detectability does not require the knowledge of the storage function V .
0 2 1
2 1
0
Proposition III.2 Let S and Ω be defined as in Proposition II.1 and define S ′ = {x ∈ X : Lm f h(x) = 0, 0 ≤ m ≤ r + n − 2}. Then, S ∩ Ω = S ′ ∩ Ω and therefore under the assumption of Proposition III.1 system (1) is Γ-detectable if S ′ ∩ Ω ⊂ Γ. The proof of the this proposition is omitted for brevity. Shiriaev’s result in Theorem II.1 states that V detectability, and hence Γ-detectability, is a sufficient condition to asymptotically stabilize Γ in the case when Γ = V −1 (0) and Γ is compact. In the more general setting under investigation, that is, when Γ ⊂ V −1 (0) and Γ is not necessarily compact, Γ-detectability is no longer a sufficient condition for asymptotic stability of Γ. We show this by means of the following counterexample. Example III.1 Consider the following system: x˙ 1 = (x22 + x23 )(−x2 ), x˙ 2 = (x22 + x23 )(x1 ), x˙ 3 = −x33 + u, with output function h(x1 , x2 , x3 ) = x33 . This system is passive with storage V = 14 x43 . On the set h−1 (0) = {(x1 , x2 , x3 ) : x3 = 0}, the system dynamics take the form: x˙ 1 = (x22 )(−x2 ), x˙ 2 = (x22 )(x1 ). If the goal set is given as Γ = {(x1 , x2 , x3 ) : x2 = x3 = 0}, then it can be seen from Figure 1 that all trajectories originating in h−1 (0) approach Γ and hence the system is Γ-detectable. Using the control u = −y, all system trajectories are bounded for all x0 ∈ R3 . Thus, for all x0 ∈ R3 , the positive limit set L+ (x0 , u) is nonempty, compact, and formed by trajectories on h−1 (0) = {x3 = 0}. For x0 ∈ / (h−1 (0) ∪ {(x1 , x2 , x3 ) : x1 = x2 = 0)}), it can be shown that the positive limit set L+ (x0 , u) is a circle which intersects Γ at equilibrium points. Thus, x(t) 6→ Γ because L+ (x0 , 0) 6⊂ Γ. This is illustrated in Figure 1. In this example the feedback u = −y guarantees that all system trajectories are bounded for all initial conditions, and
0
−1
x2
−2
Fig. 1.
−1 −2
x1
Counterexample
x(t) → h−1 (0) = V −1 (0). The Γ-detectability property guarantees that on V −1 (0), which is open-loop invariant, all trajectories approach Γ. However, these two facts do not imply that x(t) → Γ. The problem lies in the fact that, while Γ is a global attractor relative to V −1 (0), Γ is not stable relative to V −1 (0). We next introduce sufficient conditions to insure that Γ is a global attractor of the closed-loop system in the special case when all trajectories of the system are bounded. The condition, in the result below, dealing with relative stability of Γ is similar to a condition presented in [13] for equilibrium stabilization of positive systems with first integrals. Part of the proof of our result is inspired by the stability results using positive semi-definite Lyapunov functions presented in [14]. Theorem III.1 Consider system (1) and a feedback u = −ϕ(y), where ϕ : Y → U is a smooth function such that ϕ(0) = 0 and y ⊤ ϕ(y) > 0 for y 6= 0. Suppose that, for all x0 ∈ X [resp., for all x0 in a neighborhood of Γ], x(t) = φ(t, x0 , u) is bounded. If the system is Γ-detectable [resp., locally Γ-detectable] and Γ is stable with respect to the openloop system relative to V −1 (0), then Γ is a global attractor [resp., a semi-attractor] of the closed-loop system. Remark III.3 When Γ = V −1 (0), then the relative stability assumption of Theorem III.1 is trivially fulfilled.
Example III.2 Consider the following system: x˙ 1 = x2 x3 , x˙ 2 = x1 x3 − x32 , x˙ 3 = −x3 x42 + u, with output y = x3 . This system is passive with positive semi-definite storage V (x) = 1 2 2 2 x3 . Indeed, Lg V (x) = x3 and Lf V (x) = −x3 x2 ≤ 0. Consider the goal set Γ = {x : x3 = x2 = 0}. The set V −1 (0) = h−1 (0) = {x : x3 = 0} is open-loop invariant and the dynamics of the system on the set are given by x˙ 1 = 0, x˙ 2 = −x32 . Clearly, the system is Γ detectable and Γ is stable with respect to the open-loop system relative to V −1 (0). The feedback u = −y globally stabilizes Γ. IV. O UTPUT C ONVERGENCE
VS .
S ET ATTRACTIVITY
When the goal set Γ can be expressed as Γ = γ −1 (0) = {x ∈ X : γ(x) = 0}, where γ is a continuous function X → Rp , one approach to stabilizing Γ is to view γ(x) as the output of the system and pose the problem of stabilizing Γ as that of stabilizing the output. In general, however, the convergence of the output to zero is not equivalent to the
1.5
1
x2
0.5
level sets of γ 0
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x1
(a) γ(x(t)) → 0 6⇒ x(t) → γ −1 (0) 2 1.5 1
level sets of γ
0.5
x2
Proof: We prove the global result of the theorem. The proof of the local result is very similar mutatis mutandis. Since the trajectory x(t) = φ(t, x0 , u) is bounded, its positive limit set, L+ (x0 , u), is nonempty, compact, and invariant for the closed-loop system. R t Along the trajectory x(t) we have V (x(t)) − V (x0 ) ≤ 0 −y(τ )⊤ ϕ(y(τ ))dτ ≤ 0. Since V is continuous and nonincreasing, limt→∞ V (x(t)) = c ≥ 0. This implies that V (x) = c on L+ (x0 , u). Let x ¯ be a point of L+ (x0 , u) and set x ¯(t) = φ(t, x ¯, u). Since x¯(t) ∈ L+ (x0 , u), V (¯ x(t)) = c for all t ≥ 0 and 0 = V (¯ x(t)) − V (¯ x) ≤ Rthen t ⊤ −y(τ ) ϕ(y(τ ))dτ ≤ 0, so that y(t) = h(¯ x (t)) = 0, ∀t ≥ 0 0. By Γ-detectability, x ¯(t) → Γ ⊂ V −1 (0), and hence c = 0. We have thus shown that L+ (x0 , u) ⊂ V −1 (0). In order to show that x(t) → Γ we will show that, by the assumption of stability of Γ relative to V −1 (0), L+ (x0 , u) ⊂ Γ. Assume, by way of contradiction, that L+ (x0 , u) 6⊂ Γ, so that we can find a point p ∈ L+ (x0 , u) and p ∈ / Γ. Since V is positive semi-definite, it follows that for all x ∈ V −1 (0), dV (x) = 0, and hence Lg V (x) = 0, proving that V −1 (0) ⊂ h−1 (0). This fact, the invariance of L+ (x0 , u), and the fact that L+ (x0 , u) ⊂ V −1 (0) together imply that φ(t, p, u) = φ(t, p, 0) for all t ∈ R, and therefore φ(t, p, 0) ∈ L+ (x0 , u) for all t ∈ R. The invariance and closedness of L+ (x0 , u) also imply that L− (p, 0) ⊂ L+ (x0 , u) ⊂ V −1 (0). Since L− (p, 0) ⊂ V −1 (0) is open-loop invariant, by the Γ-detectability property we have that open-loop trajectories originating in L− (p, 0) are contained in L− (p, 0) and approach Γ in positive time; this fact and the closedness of L− (p, 0) imply that L− (p, 0) ∩ Γ 6= ∅. Let x ¯ be a point in L− (p, 0) ∩ Γ. Since Γ is stable with respect to the openloop system relative to V −1 (0), for any ǫ > 0 there exists a neighborhood N (Γ) such that for all z ∈ N (Γ) ∩ V −1 (0), kφ(t, z, 0)kΓ < ǫ for all t ≥ 0. Pick ǫ > 0 such that kpkΓ > ǫ. Since x¯ ∈ Γ is a negative limit point of φ(t, p, 0), there exist a point z ∈ N (Γ) and a time T > 0 such that p = φ(T, z, 0), contradicting the relative stability of Γ. Hence, L+ (x0 , u) ⊂ Γ. Since φ(t, x0 , u) → L+ (x0 , u), we conclude that φ(t, x0 , u) → Γ.
0 -0.5
Γ
-1 -1.5 -2 -1000 -800 -600 -400 -200
0
200
x1
(b) x(t) → γ −1 (0) 6⇒ γ(x(t)) → 0 Fig. 2. Counterexamples showing that output convergence is not equivalent to set attractivity.
attractivity of the goal set. This is illustrated in the next two counterexamples. Example IV.1 (γ(x(t)) → 0 6⇒ x(t) → γ −1 (0)) Consider −x2 the system x˙ 1 = 1, x˙ 2 = 1+x with output function 2 1 x2 γ(x1 , x2 ) = 1+x2 . The zero level set of this function 1 is given by γ −1 (0) = {x : x2 = 0}. The solution with initial condition (x10 , x20 ) is x1 (t) = x10 + t, x2 (t) = x20 exp(− arctan(x10 + t) + arctan x10 ). It is clear that if x20 6= 0, then x(t) 6→ γ −1 (0). The output signal is γ(x(t)) = [x20 exp(− arctan(x10 + t) + arctan x10 )]/[1 + (x10 + t)2 ]. Clearly, γ(x(t)) → 0. The problem here is that for all c 6= 0, the distance between points in γ −1 (c) and the set γ −1 (0) is not uniformly bounded; see Figure 2(a). Example IV.2 (x(t) → γ −1 (0) 6⇒ γ(x(t)) → 0) Consider 1 x2 with output function the system x˙ 1 = 1, x˙ 2 = −x 1+x21 2 2 γ(x1 (t), x2 (t)) = x2 (1 + x1 ). The zero level set of this function is the set γ −1 (0) = {x : x2 = 0}. The solution with initial condition (x10 , x20 ) is x1 (t) = x10 + t,
p x2 (t) = x20 (1 + x210 )/(t2 + 2x10 t + 1 + x210 ), so for all initial conditions, x(t) → γ −1 (0). On the other hand, γ(x1 (t), x2 (t)) = x220 [(1 + x210 )[(x10 + t)2 + 1]]/[t2 + 2x10 t + 1 + x210 ], so if x20 6= 0, γ(x1 (t), x2 (t)) 6→ 0. The problem here is that for all c 6= 0, the distance between the sets γ −1 (c) and γ −1 (0) is not bounded away from zero; see Figure 2(b). We next present necessary and sufficient conditions to establish the equivalence between output convergence and the attractivity of γ −1 (0). The proof is omitted due to space limitations. Theorem IV.1 Let γ : X → Rp be a continuous function, and let Γ = {x ∈ X : γ(x) = 0}. (i) (Class K-lower bound) A necessary and sufficient condition such that (∀{xn }n∈N ⊂ X ) (γ(xn ) → 0 =⇒ xn → Γ), is that there exists r > 0 and a class-K function α : [0, r) → R+ such that (∀kxkΓ < r) α(kxkΓ ) ≤ kγ(x)k. (ii) (Class K-upper bound) A necessary and sufficient condition such that (∀{xn }n∈N ⊂ X ) (xn → Γ =⇒ γ(xn ) → 0), is that there exists r > 0 and a class-K function β : [0, r) → R+ such that (∀kxkΓ < r) kγ(x)k ≤ β(kxkΓ ). (iii) (Bounded Sequences) For all bounded sequences {xn }n∈N ⊂ X , γ(xn ) → 0 ⇐⇒ xn → Γ. V. S ET S TABILIZATION - C ASE
OF UNBOUNDED
TRAJECTORIES
We now turn our attention to situation when trajectories are unbounded and present two results concerning the (semi) attractivity and asymptotic stability of Γ. A key ingredient here is the result in Theorem IV.1. Theorem V.1 Consider system (1) and a feedback u = −ϕ(x), where ϕ : X → U is a smooth function such that h(x)⊤ ϕ(x) ≥ ψ(kh(x)k), and ψ is a class-K function. Define λ(x) = h(x)⊤ ϕ(x). Suppose that the closed-loop system has no finite escape times and that: (i) There exists r > 0 and a class-K function α : [0, r) → R+ such that α(kxkΓ ) ≤ kh(x)k for all kxkΓ < r. (ii) There exists s ∈ (0, r] such that inf kxkΓ =s V (x) > 0. (iii) For all c > 0 there exists k ∈ R such that whenever V (x) ≤ c, either Lf λ(x) − Lg λ(x)ϕ(x) ≤ k or Lf λ(x) − Lg λ(x)ϕ(x) ≥ k. Then, Γ is a semi-attractor. If (i)-(iii) hold and (iv) there exists a class-K function β : [0, r) → R+ such that V (x) ≤ β(kxkΓ ) for all kxkΓ < r, then Γ is an attractor. Finally, if (i), (iii) hold and
(v) Γ = V −1 (0) = h−1 (0), then Γ is a global attractor. Remark V.1 Assumption (i) implies that Γ = V −1 (0)∩{x ∈ X : kxkΓ < r} = h−1 (0) ∩ {x ∈ X : kxkΓ < r}. In other words, the assumption imposes that each connected component of Γ coincides with a connected component of V −1 (0) and h−1 (0). Proof: Let x0 ∈ X be arbitrary and denote x(t) = φ(t, x0 , u). SinceR V (·) is nonnegative and t V (x(t)) − V (x(0)) ≤ − 0 λ(x(τ )) dτ ≤ 0, this implies that V (x(t)) ≤ V (x(0)) for all t ≥ 0 and Rt thus limt→∞ 0 λ(x(τ )) dτ exists and is finite. By (iii), ˙ there exists k ∈ R such that λ(x(t)) = Lf λ(x(t)) + Lg λ(x(t))ϕ(x(t)) is either ≤ k or ≥ k. Take the case where ˙ λ(x(t)) ≥ k; we will show that λ(x(t)) → 0. Suppose, by way of contradition, that λ(x(t)) 6→ 0. This implies that there exists ǫ > 0 and a sequence {tn } such that tn → ∞ and λ(x(tn )) ≥ ǫ for all n. By the mean value theorem, for all n and for all t ≥ tn , there exists sn ∈ [tn , t] such that ˙ λ(x(t)) = λ(x(tn )) + λ(x(s n ))(t − tn ) ≥ ǫ + k(t − tn ) ≥ ǫ − |k| |t − tn |. Pick ǫ0 ∈ (0, ǫ) and let δ = (ǫ − ǫ0 )/|k|. Then, for all n and for all t ∈ [tn , tn + δ], λ(t) ≥ ǫ0 > 0. Take a subsequence {tnk } such that tnk+1 − tnk ≥ δ. Since, by assumption, λ(x(t)) ≥ 0, we have Z t X X Z tnk +δ λ(x(τ ))dτ ≥ λ(x(τ ))dτ ≥ lim δǫ0 , t→∞
0
k
tnk
k
Rt
contradicting the fact that limt→∞ 0 λ(x(τ )) dτ exists and is finite. Thus, λ(x(t)) → 0 and since ψ(kh(x)k) ≤ λ(x(t)), ˙ h(x(t)) → 0. The proof for the case λ(x(t)) ≤ k is almost identical and therefore omitted. Recall (see the proof of Theorem III.1) that Γ ⊂ V −1 (0) ⊂ h−1 (0). By assumption (i), Γ = V −1 (0) ∩ {kxkΓ < r} = h−1 (0) ∩ {kxkΓ < r}. Moreover, for all kxkΓ < r, kh(x)k ≥ α(kxkΓ ) ≥ α(kxkh−1 (0) ). By (ii), c = inf kxkΓ =s V (x) is positive. Let N (Γ) = {x ∈ X : kxk < s, V (x) < c}. It follows that N (Γ) is positively invariant and thus for all x0 ∈ N (Γ), h(x(t)) → 0 and kx(t)kΓ < r. This, together with the fact that kh(x(t))k ≥ α(kx(t)kh−1 (0) ) implies that x(t) → h−1 (0)∩{kxkΓ < r} = Γ, proving that Γ is a semi-attractor. If, in addition to assumptions (i)-(iii), assumption (iv) holds then there exists δ > 0 such that {x ∈ X : kxkΓ < δ} ⊂ N (Γ)}, and thus Γ is an attractor. If assumptions (i), (iii) and (v) hold, then we have shown that (∀x0 ∈ X ) h(φ(t, x0 , u)) → 0 and by Theorem IV.1, φ(t, x0 , u) → h−1 (0) = Γ. Thus, Γ is a global attractor. By strengthening conditions (i), (ii) and removing condition (iii) in Theorem V.1, we gain stability. Theorem V.2 Consider system (1) and a feedback u = −ϕ(x), where ϕ : X → U is a smooth function such that ϕ(x)⊤ h(x) ≥ ψ(kh(x)k), and ψ is a class-K function. Suppose that the closed-loop system has no finite escape times and that:
(i) There exists r > 0 and a class-K function α : [0, r) → R+ such that α(kxkΓ ) ≤ V (x) for all kxkΓ < r. (ii) There exists a class-K function µ : [0, +∞) → R+ such that µ(V (x)) ≤ kh(x)k for all x ∈ X . Then, Γ is a stable semi-attractor of the closed-loop system. If (i), (ii) hold and (iii) there exists a class-K function β : [0, r) → R+ such that V (x) ≤ β(kxkΓ ) for all kxkΓ < r, then Γ is asymptotically stable. If (i), (ii) hold and (iv) Γ = V −1 (0) = h−1 (0), then Γ is globally attractive. Finally, if (i)-(iv) hold, then Γ is globally asymptotically stable. Remark V.2 Assuptions (i), (ii) in Theorem V.2 imply assumptions (i), (ii) of Theorem V.1. Indeed, for all kxkΓ < r, kh(x)k ≥ µ(V (x)) ≥ µ ◦ α(kxkΓ ). Moreover, the inequality V (x) ≥ α(kxkΓ ) implies inf kxkΓ =s V (x) ≥ α(s) > 0 for all s ∈ (0, r]. Proof: By assumption (ii), for all x0 ∈ X , dV (φ(t, x0 , u)) ≤ −h(φ(t, x0 , u))⊤ ϕ(φ(t, x0 , u)) dt ≤ −ψ ◦ µ(V (φ(t, x0 , u))). By the comparison lemma, it follows that V (φ(t, x0 , u)) → 0. Let ǫ ∈ (0, r), choose c > 0 such that α−1 (c) < ǫ, and define N (Γ) = {x ∈ X : V (x) < c, kxkΓ < r}. Clearly, N (Γ) is a neigbourhood of Γ. By our choice of c and assumption (i), N (Γ) ⊂ {x ∈ X : kxkΓ < ǫ) and the boundary of N (Γ) is ∂N (Γ) = {x ∈ X : V (x) = c, kxkΓ < r}. Since V is nonincreasing, it follows that N (Γ) is positively invariant and hence Γ is stable. Since V (φ(t, x0 , u)) → 0 for all x0 ∈ N (Γ), assumption (i) implies that x(t) → Γ. Hence, Γ is a semi-attractor. If, in addition, assumption (iii) holds, then there exists δ > 0 such that {x ∈ X : kxkΓ < δ} ⊂ N (Γ), and thus Γ is asymptotically stable. If assumptions (i), (ii), and (iv) hold, then we have shown that (∀x0 ∈ X ) V (φ(t, x0 , u)) → 0. Hence, by Theorem IV.1, φ(t, x0 , u) → V −1 (0) = Γ. Thus, Γ is globally attractive. If assumptions (i)-(iv) hold, then Γ is a uniformly stable global attractor, that is, Γ is globally asymptotically stable.
Example V.1 Consider the system x˙ 1 = tan−1 (x1 ), x˙ 2 = x2 (1+x21 )u, with output y = x22 . This system is passive with x22 a storage function V (x) = 12 1+x 2 . Let the goal set be Γ = 1 V −1 (0) = h−1 (0) = {x : x2 = 0}. It is seen that V does not have a class-K lower bound with respect to Γ and thus Theorem V.2 cannot be applied to this example. On the other hand, h(x) does have a class-K lower bound with respect to Γ, as kh(x)k = kxk2Γ . Moreover, kh(x)k = kxk2h−1 (0) . Using the control u = −ϕ(x), with ϕ(x) = h(x), we have Lf λ(x) − Lg λ(x)ϕ(x) = −4x62 (1 + x21 ), which is bounded
from above. Thus we have that conditions (i), (iii) and (v) of Theorem V.1 are satisfied and Γ is globally attractive. Example V.2 Consider the system x˙ 1 = 1, x˙ 2 = (1 + 0.5 cos(x21 ))u, with output y = x2 (1+0.5 cos(x21 )), which is passive with storage function V (x) = 21 x22 . Let the goal set be Γ = {x : x2 = 0} so that V −1 (0) = h−1 (0) = Γ. Moreover, V (x) = 1/2kxk2Γ and kh(x)k ≥ 0.5|x2 | = 0.5kxkΓ , so by Theorem V.2 the feedback u = −ϕ(x), with ϕ(x) = h(x), renders Γ globally asymptotically stable. It is easily seen that Theorem V.1 cannot be applied to this example because assumption (iii) does not hold. VI. C ONCLUSIONS This paper leaves several open questions that we will investigate in future work. In Theorem III.1, we assume that the trajectories of the closed-loop system are bounded. Boundedness is a property that one should seek to achieve by means of a suitable feedback. The same comment holds for Theorems V.1 and V.2, where we assume that the closed-loop system does not possess finite escape times. R EFERENCES [1] F. Albertini and E. Sontag, “Continuous control-Lyapunov functions for asymptotically controllable time-varying systems,” International Journal of. Control, vol. 72, pp. 1630–1641, 1999. [2] C. M. Kellet and A. R. Teel, “Asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function,” in Proc. of the 39th Conference on Decision and Control, (Sydney, Australia), 2000. [3] C. M. Kellet and A. R. Teel, “A converse Lyapunov theorem for weak uniform asymptotic controllability of sets,” in Proc. of the 14th MTNS, (France), 2000. [4] C. M. Kellet and A. R. Teel, “Weak converse Lyapunov theorems and control-Lyapunov functions,” SIAM Journal on Control and Optimization, vol. 42, no. 6, pp. 1934–1959, 2004. [5] A. Banaszuk and J. Hauser, “Feedback linearization of transverse dynamics for periodic orbits,” Systems and Control Letters, vol. 29, pp. 95–105, 1995. [6] C. Nielsen and M. Maggiore, “Output stabilization and maneuver regulation: A geometric approach,” Systems and Control Letters, vol. 55, pp. 418–427, 2006. [7] C. Nielsen and M. Maggiore, “Further results on transverse feedback linearization of multi-input systems,” in Proc. of the 45th IEEE Conference on Decision and Control, (San Diego, CA, USA), December 2006. [8] A. Shiriaev, “Stabilization of compact sets for passive affine nonlinear systems,” Automatica, vol. 36, pp. 1373–1379, 2000. [9] A. Shiriaev, “The notion of V -detectability and stabilization of invariant sets of nonlinear systems,” Systems and Control Letters, vol. 39, pp. 327–338, 2000. [10] A. Shiriaev and A. L. Fradkov, “Stabilization of invariant sets for nonlinear non-affine systems,” Automatica, vol. 36, pp. 1709–1715, 2000. [11] C. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence, and the global stabilization of nonlinear systems,” IEEE Transactions on Automatic Control, vol. 36, pp. 1228–1240, 1991. [12] N. P. Bathia and G. P. Szeg¨o, Dynamical Systems: Stability Theory and Applications. Berlin: Springer-Verlag, 1967. [13] P. de Leenheer and D. Aeyels, “Stabilization of positive systems with first integrals,” Automatica, vol. 38, pp. 1583 – 1589, 2002. [14] A. Iggidr, B. Kalitine, and R. Outbib, “Semidefinite Lyapunov functions stability and stabilization,” Mathematics of Control, Signals and Systems, vol. 9, pp. 95–106, 1996.
